I'm going to be a TA in an introductory course in mathematics at a technical university this fall, focusing on mathematics that the students should already be familiar with but that might need refreshing before the courses in linear algebra and calculus start.
One of the things in the course is basic factorization - both for integers and polynomials. I don't think anyone will have trouble finding the first few primes such as $2$, $3$ and $5$, but it got me thinking as to whether there is a systematic approach when trying to factor slightly higher primes such as $17$ or $31$.
Generally if I happen to be doing integer factorization I try to "feel" which prime might be possible to factor out, but that is not a very helpful thing to tell students.
Is there a better method to quickly find the factors? Obviously excluding calculators, computers and the such.
edit: Quick example exercise
Perform integer factorization on $2108$. It is obvious that $2108 = 2 * 2 * 527$.
However is there a method for quickly finding the two remaining factors, $17$ and $31$? Just by looking at $527$, I would say it is not apparent to most people which prime numbers (or even what range beyond the double digits) to start trying.